Algebraic Topology

Unit aims

The aim of the unit is to give an introduction to algebraic topology with an emphasis on cell complexes, fundamental groups and homology.

Unit description

Algebraic Topology concerns constructing and understanding topological spaces through algebraic, combinatorial and geometric techniques. In particular, groups are associated to spaces to reveal their essential structural features and to distinguish them.  In cruder terms, it is about adjectives that capture and distinguish essential features of spaces.  

The theory is powerful. We will give applications including proofs of The Fundamental Theory of Algebra and Brouwer's Fixed Point Theorem (which is important in economics).

Relation to other units

This is one of three Level M units which develop group theory in various directions. The others are Representation Theory and Galois Theory.

Learning objectives

Students should absorb the idea of algebraic invariants to distinguish between complex objects, their geometric intuition should be sharpened, they should have a better appreciation of the interconnectivity of different fields of mathematics, and they should have a keener sense of the power and applicability of abstract theories.

Transferable skills

The assimilation of abstract and novel ideas.

Geometric intuition.

How to place intuitive ideas on a rigorous footing.

Presentation skills.

Syllabus

  • Topological spaces – open sets, closed sets, product, quotient and subspace topologies, connectedness, path-connectedness, continuous maps, homeomorphisms, Hausdorff spaces
  • Homotopy
  • The definition of the fundamental group and its calculation for the circle
  • The Fundamental Theorem of Algebra
  • Brouwer's Fixed Point Theorem
  • Covering Spaces
  • van Kampen's Theorem
  • Graphs and free groups
  • Singular homology

Reading and References

W. A. Sutherland, Introduction to metric and topological spaces, Clarendon Press, Oxford.

Munkres, Topology (2nd Edition), Pearson Education

Hatcher, Algebraic Topology, Chapters 0,1,2.

O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov, N.Y.Netsvetaev,Elementary Topology

Unit code: MATHM1200
Level of study: M/7
Credit points: 20
Teaching block (weeks): 1 (1-12)
Lecturer: Professor Jeremy Rickard

Pre-requisites

MATH20006 Metric Spaces and MATH33300 Group Theory

Co-requisites

None

Methods of teaching

Lectures, problem sets and discussion of problems, student presentations. 

Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

  • 80% from coursework (problem sets)
  • 20% based on seminar presentations*

* Presentations graded on the understanding and insight they demonstrate and on the clarity, quality, lucidity and style of the delivery

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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